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arxiv: 2601.15501 · v3 · submitted 2026-01-21 · 🧮 math.RA

On orthogonality graphs of Okubo algebras

Danil Pavlinov , Svetlana Zhilina This is my paper

Pith reviewed 2026-05-16 11:34 UTC · model grok-4.3

classification 🧮 math.RA
keywords Okubo algebraorthogonality graphshortest pathsconnected componentsgraph diameterisotropic norm
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The pith

In the orthogonality graph of an Okubo algebra with isotropic norm over any field, there are at most two shortest paths between any pair of vertices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines the orthogonality graph of an Okubo algebra equipped with an isotropic norm over an arbitrary field. It identifies the connected components of the graph and determines their diameters. The work proves that any two vertices have at most two shortest paths connecting them and specifies the conditions that make such a path unique. These results clarify how orthogonality relations organize the elements of the algebra into a graph with controlled distance properties.

Core claim

The orthogonality graph of an Okubo algebra with isotropic norm over an arbitrary field has its connected components described, its diameters computed, and the property that any pair of vertices is joined by at most two shortest paths, with explicit conditions for uniqueness of that path.

What carries the argument

The orthogonality graph whose vertices are elements of the Okubo algebra and whose edges connect pairs that are orthogonal with respect to the isotropic norm.

Load-bearing premise

The Okubo algebra is equipped with an isotropic norm over an arbitrary field.

What would settle it

An explicit Okubo algebra with isotropic norm over some field F together with a pair of vertices that admits three or more distinct shortest paths would falsify the main claim.

read the original abstract

The orthogonality graph of an Okubo algebra with isotropic norm over an arbitrary field $\mathbb{F}$ is considered. Its connected components are described, and their diameters are computed. It is shown that there exist at most two shortest paths between any pair of vertices, and the conditions under which the shortest path is unique are determined.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper considers the orthogonality graph of an Okubo algebra equipped with an isotropic norm over an arbitrary field F. It describes the connected components, computes their diameters, proves that there exist at most two shortest paths between any pair of vertices, and determines the conditions under which the shortest path is unique.

Significance. If the results hold, the work supplies a precise graph-theoretic description of the multiplication structure in Okubo algebras, a class of 8-dimensional nonassociative algebras generalizing the octonions. The path-counting statements and diameter computations are concrete and hold over arbitrary fields, which strengthens their applicability to algebraic geometry and representation theory contexts.

minor comments (2)
  1. [Abstract] The abstract states the main results but does not recall the definition of an Okubo algebra or the precise meaning of the isotropic norm; a one-sentence reminder would improve accessibility.
  2. [Introduction] Notation for the bilinear form and the orthogonality relation should be fixed in a preliminary section before the statements of the diameter and path theorems.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the orthogonality graphs of Okubo algebras and for recommending minor revision. No specific major comments were raised in the report, so we have no point-by-point rebuttals to provide. We will incorporate any minor suggestions during the revision process.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives properties of the orthogonality graph (connected components, diameters, at most two shortest paths) directly from the multiplication rules and isotropic norm hypothesis of the Okubo algebra over arbitrary field F. These are standard graph-theoretic consequences of the algebra's bilinear form and orthogonality relation; no parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a uniqueness theorem that forces the result, and no ansatz is smuggled in. The derivation chain is self-contained within the given algebraic identities.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The results rest on the standard definition of Okubo algebras and the orthogonality relation induced by an isotropic norm; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Standard definition and properties of Okubo algebras over an arbitrary field
    The paper invokes the usual axioms for Okubo algebras and the isotropy condition on the norm.

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Reference graph

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